In QHD, saturation results from a balance between repulsion from the \(\omega _\mu \)  meson and attraction
from the \(\sigma \)  meson [1]}, [2]}, [3]}.
That the \(\omega _\mu \)  meson could generate a stiff EoS was first noted by
Zel'dovich [4]}. Given the coupling of the \(\omega _\mu \)  to a nucleon
\(\psi \)  as \(\sim g_\omega \, \overline{\psi } \, \omega _\mu \gamma ^\mu \, \psi \) , then at any nonzero baryon density,
\(\langle \overline{\psi } \gamma ^0 \psi \rangle = n_B \ne 0\) , a condensate for \(\omega _0\)  is automatically generated
As a vector field, an expectation value for \(\omega _0\)  is generated whenever there is net baryon charge in the system, and is also why
the energy density increases. This is exactly like QED with net charge, except that here the \(\omega _\mu \)  meson is massive.
In contrast, for the chiral condensate the energy density is lowered when it has a nonzero expectation value.
I thank L. McLerran for discussions on this point.:
\({\cal L}_\omega ^B = - g_\omega n_B \omega _0 + \frac{m_\omega ^2 \omega _\mu ^2}{2} \Rightarrow \langle \omega _0 \rangle = \frac{g_\omega }{m_\omega ^2} \, n_B \; .\) 
